Question

If $A=\left[\begin{array}{ccc}3& 0& 0\\ 2& 2& 0\\ 4& 5& 3\end{array}\right],$ then which of the following is (are) true?

• A. $A$ is invertible
• B. $tr\left(adj\right(adj\left(A\right)\left)\right)=144$
• C. $tr\left(adj\right(adj\left(A\right)\left)\right)=108$
• D. $|adj\left(A\right)|$ is less than $400$

Answer 1

Answer: (A), (B), (D)

$|adj\left(A\right)|$ is less than $400$

$A=\left[\begin{array}{ccc}3& 0& 0\\ 2& 2& 0\\ 4& 5& 3\end{array}\right]$
$|A|=18\ne 0$
$\therefore A$ is invertible.


We know that $A(adjA)=|A|I_n$
Replace $A$ by $adjA,$ we have
$(adjA\left)(adj(adjA\right))=|(adjA)|I_n$
$\Rightarrow (adjA\left)(adj(adjA\right))=|A{|}^{n-1}{I}_n$
$\Rightarrow (adj(adjA\left)\right)=|A{|}^{n-2}A$
Since $n=3,$ (order of matrix)
$\therefore (adj(adjA\left)\right)=|A|A=18A$

$tr\left(adj\right(adjA\left)\right)=18tr\left(A\right)=18\times 8$
$\therefore tr\left(adj\right(adjA\left)\right)=144$


$|adjA|=|A{|}^2={18}^2=324<400$